The alternating algorithm in a uniformly convex and uniformly smooth Banach space

Let X   be a uniformly convex and uniformly smooth Banach space. Assume that the MiMi, i=1,…,ri=1,…,r, are closed linear subspaces of X  , PMiPMi is the best approximation operator to the linear subspace MiMi, and M:=M1+⋯+MrM:=M1+⋯+Mr. We prove that if M is closed, then the alternating algorithm given by repeated iterations of(I−PMr)(I−PMr−1)⋯(I−PM1)(I−PMr)(I−PMr−1)⋯(I−PM1) applied to any x∈Xx∈X converges to x−PMxx−PMx, where PMPM is the best approximation operator to the linear subspace M  . This result, in the case r=2r=2, was proven in Deutsch [4].